Estimating Fidelity to a Reference Quantum State

2026-06-24

AI Summary

This work investigates the sample complexity of estimating the fidelity between an unknown quantum state and a known reference state up to an additive error ε, with a focus on scenarios where either the reference state or the unknown state exhibits low-rank structure. By integrating tools from quantum information theory, statistical estimation, and low-rank assumptions, the authors improve the sample complexity from O(r²log²(1/ε)/ε⁴) to the optimal O(r²/ε²) when the reference state has rank r, and establish a matching lower bound of Ω(r/ε²). The analysis also extends to the setting where the unknown state is low-rank while the reference state is arbitrary. These results yield nearly tight sample complexity bounds across multiple regimes and significantly broaden the theoretical framework for fault-tolerant quantum state certification.

Abstract

We consider the problem of estimating the fidelity of an unknown quantum state to a known reference state to within additive error $\varepsilon$. We show that the sample complexity is $O(r^2/\varepsilon^2)$ with optimal $\varepsilon$-dependence when the reference state is of rank $r$, improving the previous best $O(r^2\log^2(1/\varepsilon)/\varepsilon^4)$ due to Utsumi, Nakata, Wang, and Takagi (QIP 2026). We also provide a lower bound of $Ω(r/\varepsilon^2)$, improving the previous best $Ω(r/\varepsilon+1/\varepsilon^2)$, with implications to quantum query complexity. Moreover, we further consider the case where the unknown state is of rank at most $r$ while the reference state can be arbitrary, for which the sample complexity is shown to be $O(r^2/\varepsilon^4)$. As an application, we present an approach to tolerant quantum state certification, generalizing the exact certification studied in Bădescu, O'Donnell, and Wright (STOC 2019).

Problem

• quantum state fidelity
• sample complexity
• quantum state certification
• additive error
• rank-constrained quantum states

Innovation

• quantum fidelity estimation
• sample complexity
• rank-constrained quantum states
• quantum state certification
• quantum query complexity